Question

What is the product of \[ 1.11\times2^3 \] and \[ 1.0101\times2^6 ? \]

• A. \(1.0010011\times2^{10}\)
• B. \(1.0010011\times2^{9}\)
• C. \(11.1101\times2^{18}\)
• D. \(11.1101\times2^{9}\)

Hint:

For floating-point multiplication: \[ \text{Mantissas multiply} \] \[ \text{Exponents add} \] and finally normalize the answer so that only one non-zero digit appears before the binary point.

Answer 1

The Correct Option is A

Concept: When multiplying floating-point binary numbers:
• Multiply the mantissas.
• Add the exponents.
• Normalize the result if necessary. Thus, \[ (a\times2^m)(b\times2^n) = (ab)\times2^{m+n}. \]

Step 1: Add the exponents.
Given exponents: \[ 3 \] and \[ 6. \] Therefore, \[ 2^3\times2^6=2^9. \]

Step 2: Multiply the mantissas.
\[ 1.11_2 = 1+\frac12+\frac14 = 1.75. \] \[ 1.0101_2 = 1+\frac14+\frac1{16} = 1.3125. \] Thus \[ 1.75\times1.3125 = 2.296875. \]

Step 3: Convert back to binary.
\[ 2.296875 = 10.010011_2. \] Hence, \[ 10.010011_2\times2^9. \]

Step 4: Normalize the result.
\[ 10.010011_2 = 1.0010011_2\times2^1. \] Therefore, \[ 1.0010011\times2^{10}. \]

Step 5: Write the final answer.
\[ \boxed{1.0010011\times2^{10}} \] Hence option (A) is correct.